| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 2·13-s − 15-s − 6·17-s − 4·19-s − 21-s + 8·23-s + 25-s + 27-s − 2·29-s − 8·31-s + 35-s − 2·37-s + 2·39-s − 6·41-s − 45-s + 49-s − 6·51-s + 6·53-s − 4·57-s − 4·59-s − 6·61-s − 63-s − 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s − 1.45·17-s − 0.917·19-s − 0.218·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.169·35-s − 0.328·37-s + 0.320·39-s − 0.937·41-s − 0.149·45-s + 1/7·49-s − 0.840·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s − 0.768·61-s − 0.125·63-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.491464015446977178115614904839, −7.40257787864273811255199237969, −6.91027325588170511486091336888, −6.13838740832347652910249824981, −5.05780339959144972924105777872, −4.23091664956142410509125230141, −3.50572166183769855306341680149, −2.63443073981186150263482763066, −1.57811420584642325475547468966, 0,
1.57811420584642325475547468966, 2.63443073981186150263482763066, 3.50572166183769855306341680149, 4.23091664956142410509125230141, 5.05780339959144972924105777872, 6.13838740832347652910249824981, 6.91027325588170511486091336888, 7.40257787864273811255199237969, 8.491464015446977178115614904839
